1. Tardigrade
  2. Question
  3. Mathematics
  4. Consider the function: f (x)=[x]+|1-x|, -1 le x le3 where [x] is the greatest integer function. Statement 1: f is not continuous at x = 0, 1,2 and 3. Statement 2: f(x) = begincases -x, -1 le x < 0 1-x 0 le x < 1 1 + x 1 le x < 2 2 + x &2 le x le 3 endcases

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Q. Consider the function:
$f \left(x\right)=\left[x\right]+\left|1-x\right|, -1 \le x \le3$ where $\left[x\right]$ is the greatest integer function.
Statement 1: $f$ is not continuous at $x = 0, 1,2$ and $3$.
Statement 2 : $f(x) = \begin{cases} -x, & -1 \le x < 0 \\ 1-x & 0 \le x < 1\\ 1 + x & 1 \le x < 2 \\ 2 + x &2\le x \le 3 \end{cases} $

JEE MainJEE Main 2013Continuity and Differentiability

Solution:

Let $f \left(x\right)=\left[x\right]+\left|1-x\right|, -1 \le x \le3$
where $\left[x\right]=$ greatest integer function.
$f$ is not continuous at $x = 0,1,2,3$
But in statement-2 $f (x)$ is continuous at $x = 3$.
Hence, statement-$1$ is true and $2$ is false.